Chapter 13, Problem 10STP

To determine

The number of ways to arrange the first $3$ batters by Coach.

Answer to Problem 10STP

The number of ways to arrange the first $3$ batters by Coach are $84$

Explanation of Solution

Given:

  $\begin{array}{l}n=9\\ r=3\end{array}$

Concept Used:

Combination of choosing $r$ unique ways from $r$ possibilities is given by the following expression:

  $C{}_r=\frac{n!}{r!\left(n-r\right)!}$

Calculation:

Combination of arrange the first $3$ batters from $9$ batters possibilities is given by the following expression:

  $\begin{array}{l}C{}_r=\frac{n!}{r!\left(n-r\right)!}\\ C{}_3=\frac{9!}{3!\left(9-3\right)!}\\ C{}_3=\frac{9!}{3!6!}\\ C{}_3=\frac{9\times 8\times 7\times 6!}{\left(3\times 2\right)6!}\\ C{}_3=\frac{9\times 8\times 7}{\left(3\times 2\right)}\\ C{}_3=3\times 4\times 7\\ C{}_3=84\end{array}$

Through the $84$ ways, she can arrange the first $3$ batters

Conclusion:

Hence, the number of ways to arrange the first $3$ batters by Coach are $C{}_3=84$ .